implement tracking through a quadrupole

[dtabell]

element Quadrupole

• without spin
• with spin

[dtabell]

@mattsignorelli I have an initial implementation. see the branch 5-track-symplectic-quad.
It needs testing, and I have plenty of questions about both type stability and performance.
The former we should address ASAP. The latter we can address later.

In addition, I added two new functions to utils.jl: sincu and sinch.
These also need attention from an expert in type stability.

We should also discuss spin.

[dtabell]

Some possible tests:

• zero strength quad == drift
• low-energy beam
• high-energy beam
• ultra-relativistic beam
• $e^-$, $e^+$, $p^+$, $p^-$
• symmetry checks, e.g. $\mathcal{M}_Q(-x,0,-y,0,\delta)$
• $+q \text{ in } QD \equiv -q \text{ in } QF$
• very-short, very-strong quad should look like a thin lens
• $\mathcal{M}_Q(L) \mathcal{M}_Q(L)$ should yield very nearly the same result as $\mathcal{M}_Q(2L)$
• the previous comparison should be exact if we use two slices for the latter mapping
• particles with $q$ some multiple of the elementary charge $e$
• others ???

[dtabell]

The (non-spin) quad is done, and low energy beam tests are done.

[dtabell]

Have tests using electrons at energies 5 keV, 1 MeV, and 1 GeV,
and protons at 250 GeV, The electron tests all pass. Still working
on the proton tests.

[dtabell]

After I updated to use AtomicAndPhysicalConstants.jl, proton tests now pass.

[dtabell]

I think the addition of spin tracking deserves a separate issue.
So ... closing this one as complete.